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Optimizing Perfectly Matched Layers in Discrete Contexts
http://hdl.handle.net/2268/165708
Title: Optimizing Perfectly Matched Layers in Discrete Contexts
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<br/>Author, coauthor: Modave, Axel; Delhez, Eric; Geuzaine, Christophe
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<br/>Abstract: Perfectly Matched Layers (PMLs) are widely used for the numerical simulation of wavelike problems defined on large or infinite spatial domains. However, for both the timedependent and the timeharmonic cases, their performance critically depends on the socalled absorption function. This paper deals with the choice of this function when classical numerical methods are used (based on finite differences, finite volumes, continuous finite elements and discontinuous finite elements). After reviewing the properties of the PMLs at the continuous level, we analyse how they are altered by the different spatial discretizations. In the light of these results, different shapes of absorption function are optimized and compared by means of both one and twodimensional representative timedependent cases. This study highlights the advantages of the socalled shifted hyperbolic function, which is efficient in all cases and does not require the tuning of a free parameter, by contrast with the widely used polynomial functions.
Wed, 16 Apr 2014 10:10:17 GMT

Generalized Pointwise Hölder Spaces
http://hdl.handle.net/2268/165274
Title: Generalized Pointwise Hölder Spaces
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<br/>Author, coauthor: Nicolay, Samuel; Kreit, Damien
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<br/>Abstract: In [8,7], the properties of generalized uniform Hölder spaces have been investigated. The idea underlying the definition is to replace the exponent α of the usual spaces Λ^α(R^d) (see e.g. [6]) with a sequence σ satisfying some conditions. The soobtained spaces Λ^σ(R^d) generalize the spaces Λ^α(R^d); the spaces Λ^σ(R^d) are actually the spaces B^{1/σ_{∞,∞}(R^d), but they present specific properties (induced by L^∞norms) when compared to the more general spaces B^{1/σ}_{p,q}(R^d) studied in [2,4,1,5,9,10] for example. Indeed it is shown in [8,7] that most of the usual properties holding for the spaces Λ^α(R^d) can be transposed to the spaces Λ^σ(R^d).
Here, we introduce the pointwise version of these spaces: the spaces Λ^{σ,M}(x_0), with x_0∈R^d. Let us recall that a function f∈L^∞_loc(R^d) belongs to the usual pointwise Hölder space Λ^α(x_0) (α>0) if and only if there exist C,J>0 and a polynomial P of degree at most α such that
sup_{h≤2^{−j}} f(x_0+h)−P(h)≤C2^{−jα}.
As in [8,7], the idea is again to replace the sequence (2^{−jα})_j appearing in this inequality with a positive sequence (σ_j)j such that σ_{j+1}/σ_j is bounded (for any j); the number M stands for the maximal degree of the polynomial (this degree can not be induced by a sequence σ). By doing so, one tries to get a better characterization of the regularity of the studied function f. Generalizations of the pointwise Hölder spaces have already been proposed (see e.g. [3]), but, to our knowledge, the definition we give here is the most general version and leads to the sharpest results. [1] Alexandre Almeida. Wavelet bases in generalized Besov spaces. J. Math. Anal. Appl., 304(1):198–211, 2005.
[2] António M. Caetano and Susana D. Moura. Local growth envelopes of spaces of generalized smoothness: the critical case. Math. Inequal. Appl., 7(4):573–606, 2004.
[3] Marianne Clausel. Quelques notions d'irrégularité uniforme et ponctuelle : le point de vue ondelettes. PhD thesis, University of Paris XII, 2008.
[4] Walter Farkas. Function spaces of generalised smoothness and pseudodifferential operators associated to a continuous negative definite function. Habilitation Thesis, 2002.
[5] Walter Farkas and HansGerd Leopold. Characterisations of function spaces of generalised smoothness. Ann. Mat. Pura Appl., IV. Ser., 185(1):1–62, 2006.
[6] Steven G. Krantz. Lipschitz spaces, smoothness of functions, and approximation theory. Exposition. Math., 1(3):193–260, 1983.
[7] Damien Kreit and Samuel Nicolay. Characterizations of the elements of generalized HölderZygmund spaces by means of their representation. J. Approx. Theory, to appear, 10.1016/j.jat.2013.04.003.
[8] Damien Kreit and Samuel Nicolay. Some characterizations of generalized Hölder spaces. Math. Nachr., 285(1718):2157–2172, 2012.
[9] Thomas Kühn, HansGerd Leopold, Winfried Sickel, and Leszek Skrzypczak. Entropy numbers of embeddings of weighted Besov spaces II. Proceedings of the Edinburgh Mathematical Society (Series 2), 49(02):331–359, 2006.
[10] Susana D. Moura. On some characterizations of Besov spaces of generalized smoothness. Math. Nachr., 280(910):1190–1199, 2007.
Mon, 07 Apr 2014 15:11:43 GMT

Contraction of monotone phasecoupled oscillators
http://hdl.handle.net/2268/165110
Title: Contraction of monotone phasecoupled oscillators
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<br/>Author, coauthor: Mauroy, Alexandre; Sepulchre, Rodolphe
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<br/>Abstract: This paper establishes a global contraction property for networks of phasecoupled oscillators characterized by a monotone coupling function. The contraction measure is a total variation distance. The contraction property determines the asymptotic behavior of the network, which is either finitetime synchronization or asymptotic convergence to a splay state. © 2012 Elsevier B.V. All rights reserved.
Thu, 03 Apr 2014 06:35:36 GMT

Local stability results for the collective behaviors of infinite populations of pulsecoupled oscillators
http://hdl.handle.net/2268/165109
Title: Local stability results for the collective behaviors of infinite populations of pulsecoupled oscillators
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<br/>Author, coauthor: Mauroy, Alexandre; Sepulchre, Rodolphe
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<br/>Abstract: In this paper, we investigate the behavior of pulsecoupled integrateandfire oscillators. Because the stability analysis of finite populations is intricate, we investigate stability results in the approximation of infinite populations. In addition to recovering known stability results of finite populations, we also obtain new stability results for infinite populations. In particular, under a weak coupling assumption, we solve for the continuum model a conjecture still prevailing in the finite dimensional case. © 2011 IEEE.
Thu, 03 Apr 2014 06:35:02 GMT

Global analysis of a continuum model for monotone pulsecoupled oscillators
http://hdl.handle.net/2268/165108
Title: Global analysis of a continuum model for monotone pulsecoupled oscillators
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<br/>Author, coauthor: Mauroy, Alexandre; Sepulchre, Rodolphe
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<br/>Abstract: We consider a continuum of phase oscillators on the circle interacting through an impulsive instantaneous coupling. In contrast with previous studies on related pulsecoupledmodels, the stability results obtained in the continuum limit are global. For the nonlinear transport equation governing the evolution of the oscillators, we propose (under technical assumptions) a global Lyapunov function which is induced by a total variation distance between quantile densities. The monotone time evolution of the Lyapunov function completely characterizes the dichotomic behavior of the oscillators: either the oscillators converge in finite time to a synchronous state or they asymptotically converge to an asynchronous state uniformly spread on the circle. The results of the present paper apply to popular phase oscillators models (e.g., the wellknown leaky integrateandfire model) and show a strong parallel between the analysis of finite and infinite populations. In addition, they provide a novel approach for the (global) analysis of pulsecoupled oscillators. © 2012 IEEE.
Thu, 03 Apr 2014 06:34:11 GMT

Global Isochrons and Phase Sensitivity of Bursting Neurons
http://hdl.handle.net/2268/165076
Title: Global Isochrons and Phase Sensitivity of Bursting Neurons
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<br/>Author, coauthor: Mauroy, Alexandre; Rhoads, Blane; Moehlis, Jeff; Mezic, Igor
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<br/>Abstract: Phase sensitivity analysis is a powerful method for studying (asymptotically periodic) bursting neuron models. One popular way of capturing phase sensitivity is through the computation of isochronssubsets of the state space that each converge to the same trajectory on the limit cycle. However, the computation of isochrons is notoriously difficult, especially for bursting neuron models. In [W. E. Sherwood and J. Guckenheimer, SIAM J. Appl. Dyn. Syst., 9 (2010), pp. 659703], the phase sensitivity of the bursting HindmarshRose model is studied through the use of singular perturbation theory: cross sections of the isochrons of the full system are approximated by those of fast subsystems. In this paper, we complement the previous study, providing a detailed phase sensitivity analysis of the full (threedimensional) system, including computations of the full (twodimensional) isochrons. To our knowledge, this is the first such computation for a bursting neuron model. This was made possible thanks to the numerical method recently proposed in [A. Mauroy and I. Mezić, Chaos, 22 (2012), 033112]relying on the spectral properties of the socalled Koopman operatorwhich is complemented with the use of adaptive quadtree and octree grids. The main result of the paper is to highlight the existence of a region of high phase sensitivity called the almost phaseless set and to completely characterize its geometry. In particular, our study reveals the existence of a subset of the almost phaseless set that is not predicted by singular perturbation theory (i.e., by the isochrons of fast subsystems). We also discuss how the almost phaseless set is related to empirically observed phenomena such as addition/deletion of spikes and to extrema of the phase response of the system. Finally, through the same numerical method, we show that an elliptic bursting model is characterized by a very high phase sensitivity and other remarkable properties.
Wed, 02 Apr 2014 16:53:05 GMT

A spectral operatortheoretic framework for global stability
http://hdl.handle.net/2268/165075
Title: A spectral operatortheoretic framework for global stability
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<br/>Author, coauthor: Mauroy, Alexandre; Mezic, Igor
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<br/>Abstract: The global description of a nonlinear system through the linear Koopman operator leads to an efficient approach to global stability analysis. In the context of stability analysis, not much attention has been paid to the use of spectral properties of the operator. This paper provides new results on the relationship between the global stability properties of the system and the spectral properties of the Koopman operator. In particular, the results show that specific eigenfunctions capture the system stability and can be used to recover known notions of classical stability theory (e.g. Lyapunov functions, contracting metrics). Finally, a numerical method is proposed for the global stability analysis of a fixed point and is illustrated with several examples.
Wed, 02 Apr 2014 16:42:02 GMT

Isostables, isochrons, and Koopman spectrum for the actionangle representation of stable fixed point dynamics
http://hdl.handle.net/2268/165074
Title: Isostables, isochrons, and Koopman spectrum for the actionangle representation of stable fixed point dynamics
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<br/>Author, coauthor: Mauroy, Alexandre; Mezić, Igor; Moehlis, Jeff
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<br/>Abstract: For asymptotically periodic systems, a powerful (phase) reduction of the dynamics is obtained by computing the socalled isochrons, i.e. the sets of points that converge toward the same trajectory on the limit cycle. Motivated by the analysis of excitable systems, a similar reduction has been attempted for nonperiodic systems admitting a stable fixed point. In this case, the isochrons can still be defined but they do not capture the asymptotic behavior of the trajectories. Instead, the sets of interest  that we call " isostables"  are defined in the literature as the sets of points that converge toward the same trajectory on a stable slow manifold of the fixed point. However, it turns out that this definition of the isostables holds only for systems with slowfast dynamics. Also, efficient methods for computing the isostables are missing. The present paper provides a general framework for the definition and the computation of the isostables of stable fixed points, which is based on the spectral properties of the socalled Koopman operator. More precisely, the isostables are defined as the level sets of a particular eigenfunction of the Koopman operator. Through this approach, the isostables are unique and welldefined objects related to the asymptotic properties of the system. Also, the framework reveals that the isostables and the isochrons are two different but complementary notions which define a set of actionangle coordinates for the dynamics. In addition, an efficient algorithm for computing the isostables is obtained, which relies on the evaluation of Laplace averages along the trajectories. The method is illustrated with the excitable FitzHughNagumo model and with the Lorenz model. Finally, we discuss how these methods based on the Koopman operator framework relate to the global linearization of the system and to the derivation of special Lyapunov functions. © 2013 Elsevier B.V. All rights reserved.
Wed, 02 Apr 2014 16:16:34 GMT

On the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics
http://hdl.handle.net/2268/165073
Title: On the use of Fourier averages to compute the global isochrons of (quasi)periodic dynamics
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<br/>Author, coauthor: Mauroy, Alexandre; Mezić, Igor
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<br/>Abstract: The concept of isochrons is crucial for the analysis of asymptotically periodic systems. Roughly, isochrons are sets of points that partition the basin of attraction of a limit cycle according to the asymptotic behavior of the trajectories. The computation of global isochrons (in the whole basin of attraction) is however difficult, and the existing methods are inefficient in highdimensional spaces. In this context, we present a novel (forward integration) algorithm for computing the global isochrons of highdimensional dynamics, which is based on the notion of Fourier time averages evaluated along the trajectories. Such Fourier averages in fact produce eigenfunctions of the Koopman semigroup associated with the system, and isochrons are obtained as level sets of those eigenfunctions. The method is supported by theoretical results and validated by several examples of increasing complexity, including the 4dimensional HodgkinHuxley model. In addition, the framework is naturally extended to the study of quasiperiodic systems and motivates the definition of generalized isochrons of the torus. This situation is illustrated in the case of two coupled Van der Pol oscillators. © 2012 American Institute of Physics.
Wed, 02 Apr 2014 16:12:51 GMT

A Weak Local Irregularity Property in $S^\nu$ spaces
http://hdl.handle.net/2268/164918
Title: A Weak Local Irregularity Property in $S^\nu$ spaces
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<br/>Author, coauthor: Clausel, Marianne; Nicolay, Samuel
Mon, 31 Mar 2014 16:20:13 GMT

A new multifractal formalism based on wavelet leaders : detection of non concave and non increasing spectra (Part I)
http://hdl.handle.net/2268/164892
Title: A new multifractal formalism based on wavelet leaders : detection of non concave and non increasing spectra (Part I)
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<br/>Author, coauthor: Esser, Céline; Kleyntssens, Thomas; Nicolay, Samuel; Bastin, Françoise
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<br/>Abstract: Multifractal analysis is concerned with the study of very irregular signals. For such functions, the pointwise regularity may change widely from a point to another. Therefore, it is more interesting to determine the spectrum of singularities of the signal, which is the Hausdorff dimension of the
set of points which have the same Hölder exponent. The spectrum of singularities of many mathematical functions can be determined directly from its definition. However, for many reallife signals, the numerical determination of their Hölder regularity is not feasible. Therefore, one cannot expect to have a direct access to their spectrum of singularities and one has to find an indirect way to compute it. A multifractal formalism is a formula which is expected to yield the spectrum of singularities from quantities which are numerically computable. Several multifractal formalisms based on the wavelet coefficients of a signal have been proposed to estimate its spectrum. The most widespread of these formulas is the socalled thermodynamic multifractal formalism, based on the FrishParisi conjecture. This formalism presents two drawbacks: it can hold only for spectra that are concave and it can yield only the increasing part of the spectrum. This first problem can be avoided using Snu spaces. The second one can be avoided using a formalism based on wavelet leaders of the signal.
In this talk, we propose a new multifractal formalism, based on a generalization of the Snu spaces using wavelet leaders. It allows to detect non concave and non increasing spectra. An implementation of this method is presented in the talk "A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part II)" of T. Kleyntssens.
Mon, 31 Mar 2014 14:32:10 GMT

A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part II)
http://hdl.handle.net/2268/164867
Title: A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part II)
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<br/>Author, coauthor: Kleyntssens, Thomas; Esser, Céline; Nicolay, Samuel
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<br/>Abstract: This talk follows "A new multifractal formalism based on wavelet leaders: detection of non concave and non increasing spectra (Part I)" given by Céline Esser. For reallife signals, it is impossible to compute the spectrum of singularities by using its definition. A multifractal formalism is used to approximate this spectrum.
We present a new multifractal formalism for non concave and non increasing spectra based on wavelet leaders. In this talk, an implementation of this formalism is given and several numerical examples are presented.
Mon, 31 Mar 2014 07:19:36 GMT

About the Multifractal Nature of Cantor's Bijection
http://hdl.handle.net/2268/164643
Title: About the Multifractal Nature of Cantor's Bijection
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<br/>Author, coauthor: Simons, Laurent; Nicolay, Samuel
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<br/>Abstract: In this talk, we present the Cantor's bijection between the irrational numbers of the unit interval [0,1] and the irrational numbers of the unit square [0,1]². We explore the regularity and the fractal nature of this map. This talk is based on a joint work with S. Nicolay.
Thu, 27 Mar 2014 08:40:02 GMT

Mémoire sur les courbes du troisième ordre (seconde partie)
http://hdl.handle.net/2268/164471
Title: Mémoire sur les courbes du troisième ordre (seconde partie)
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<br/>Author, coauthor: Folie, François; Le Paige, Constantin
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<br/>Abstract: In the first part of the thesis, the authors describe successively theories of homography, involution and third order crossratio. In this one, they apply it to the cubics.; Dans la première partie du mémoire, les auteurs exposent successivement les théories de l’homographie, de l’involution et du rapport anharmonique du troisième ordre. Dans celleci, ils en font l’application aux cubiques.
Fri, 21 Mar 2014 18:44:19 GMT

dilemmas in cluster analysis
http://hdl.handle.net/2268/164465
Title: dilemmas in cluster analysis
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<br/>Author, coauthor: Sauvageot, Nicolas
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<br/>Abstract: Cluster analysis is a set of multivariate procedures to detect natural groupings in data. The objective of those methods is to group a set of objects in such a way that objects in the same group (called cluster) are more similar to each other than to those in other groups. Organizing data into sensible groupings arises naturally in many scientific fields as psychology, biology, statistics, bioinformatics, marketing, and so on.
However, the obtained solution is not unique and it strongly depends upon the analyst’s choices. Representation and normalization scheme, selection of distance measures and a clustering algorithm, choice of the number of clusters and their interpretations are all subjective choices which change the final output. Those decisions are mainly guided by the purpose of grouping, domain knowledge and the individual data set. Therefore, cluster validity assessment should be performed to evaluate the validity of the obtained clusters and to find the partitioning that best fits the underlying data.
I provide a brief overview of clustering, summarize well known algorithms, and discuss the major challenges and key issues in performing clustering analysis.
Fri, 21 Mar 2014 14:10:22 GMT

SECOND ORDER SYMMETRIES OF THE CONFORMAL LAPLACIAN
http://hdl.handle.net/2268/164312
Title: SECOND ORDER SYMMETRIES OF THE CONFORMAL LAPLACIAN
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<br/>Author, coauthor: Michel, JeanPhilippe; Radoux, Fabian; Silhan, Josef
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<br/>Abstract: Let (M,g) be an arbitrary pseudoRiemannian manifold of dimension at least 3. We determine the form of all the conformal symmetries of the conformal (or Yamabe) Laplacian on (M,g), which are given by differential operators of second order. They are constructed from conformal Killing 2tensors satisfying a natural and conformally invariant condition. As a consequence, we get also the classification of the second order symmetries of the conformal Laplacian. Our results generalize the ones of Eastwood and Carter, which hold on conformally flat and Einstein manifolds respectively. We illustrate our results on two families of examples in dimension three.
Mon, 17 Mar 2014 05:33:16 GMT

Mémoire sur les courbes du troisième ordre (première partie)
http://hdl.handle.net/2268/164251
Title: Mémoire sur les courbes du troisième ordre (première partie)
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<br/>Author, coauthor: Folie, François; Le Paige, Constantin
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<br/>Abstract: The authors describe successively theories of homography, involution and crossratio of third order.; Les auteurs exposent successivement les théories de l’homographie, de l’involution et du rapport anharmonique du troisième ordre.
Sat, 15 Mar 2014 09:26:04 GMT

Fondements d'une géométrie supérieure cartésienne
http://hdl.handle.net/2268/164214
Title: Fondements d'une géométrie supérieure cartésienne
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<br/>Author, coauthor: Folie, François
Thu, 13 Mar 2014 18:26:32 GMT

The SchwarzChristoffel transformation as a tool in applied probability
http://hdl.handle.net/2268/164213
Title: The SchwarzChristoffel transformation as a tool in applied probability
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<br/>Author, coauthor: Swan, Yvik; Bruss, F. Thomas
Thu, 13 Mar 2014 16:40:37 GMT

Essai sur une exposition nouvelle de la théorie analytique des probabilités à postériori
http://hdl.handle.net/2268/164178
Title: Essai sur une exposition nouvelle de la théorie analytique des probabilités à postériori
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<br/>Author, coauthor: Meyer, Antoine; Folie, François
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<br/>Abstract: The author wants to make more accurate the probability calculations and to concentrate the methods and the principles in the presentation of the probabilities theory. First, he puts no doubt on the series convergence by means of three demonstrated theorems. Then he resorts to an only process, the Fourier’s and LejeuneDirichlet’s theorems, for the putting in equation. Finally, he highlights the dependence of the special cases of the probabilities theory due to Laplace.; L’auteur veut rendre plus rigoureux les calculs des probabilités et concentrer les méthodes et les principes dans l’exposition de la théorie des probabilités à postériori. Il met d’abord hors de doute la convergence des séries à l’aide de trois théorèmes démontrés. Il a recours ensuite à un procédé unique, les théorèmes de Fourier et de LejeuneDirichlet, pour la mise en équation. Enfin, il fait ressortir la dépendance des cas spéciaux de la théorie des probabilités à postériori due à Laplace.
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<br/>Commentary: Ouvrage terminé d'après les manuscrits d'Antoine Meyer
Wed, 12 Mar 2014 19:49:36 GMT